DNF Sparsification and a Faster Deterministic Counting Algorithm

Parikshit Gopala; Raghu Meka; Omer Reingold

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arXiv · 1205.3534

DNF Sparsification and a Faster Deterministic Counting Algorithm

Parikshit Gopala·Raghu Meka·Omer Reingold

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Abstract

Given a DNF formula on n variables, the two natural size measures are the number of terms or size s(f), and the maximum width of a term w(f). It is folklore that short DNF formulas can be made narrow. We prove a converse, showing that narrow formulas can be sparsified. More precisely, any width w DNF irrespective of its size can be $\epsilon$-approximated by a width $w$ DNF with at most $(w\log(1/\epsilon))^{O(w)}$ terms. We combine our sparsification result with the work of Luby and Velikovic to give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF. Given a formula on n variables with poly(n) terms, we give a deterministic $n^{\tilde{O}(\log \log(n))}$ time algorithm that computes an additive $\epsilon$ approximation to the fraction of satisfying assignments of f for $\epsilon = 1/\poly(\log n)$. The previous best result due to Luby and Velickovic from nearly two decades ago had a run-time of $n^{\exp(O(\sqrt{\log \log n}))}$.

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BibTeX RIS

Parikshit Gopala, Raghu Meka, Omer Reingold. 2012-05-16. DNF Sparsification and a Faster Deterministic Counting Algorithm. https://arxiv.org/abs/1205.3534

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